Journal of Applied Probability & Statistics Vol. 5, No. x, xxx xxx c 2010 Dixie W Publishing Corporation, U. S. A.
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1 Jounal of Applied Pobability & Statistics Vol. 5, No. x, xxx xxx c 21 Dixie W Publishing Copoation, U. S. A. Bounding the Ruin Pobability unde Inteest Foce Kume Pial Das Depatment of Mathematics, Lama Univesity, Beaumont, Texas-7771, U.S.A. pialdas@gmail.com Abstact Fo the uin poblem unde inteest foce the Dubin s appoach fo matingale has been used to obtain an uppe bound on the pobability of uin. The obtained estimate fo the uin pobability in the compound Poisson isk model is appopiate when the adjustment coefficient of the individual claim size distibution does not exist. This uppe bound is easy to use and does equie less igoous moment condition. Moeove, this estimate consides that the foce of inteest is nonzeo. At least fo few distibutions this uppe bound is tighte than the uppe bound found elsewhee. Keywods: Suplus pocess, uin pobability, Matingale, Camé-Lundbeg condition, paeto distibution. 21 Mathematics Subject Classification: 6G44,62P5. 1 Intoduction Often in examining the natue of the isk associated with a potfolio of business, one uns into difficulty in quantifying this isk even if one has modeled the distibution of total claims. In addition, it s of inteest to assess how the potfolio may be expected to pefom ove an extended peiod of time. One appoach to answeing these and othe questions concens the use of uin theoy (Panje,1992). The uin poblem fo a compound Poisson isk model has been consideed by seveal authos. Gebe, Goovaets, and Kaas (1987) consideed the pobability that uin occus with initial suplus u and foce of inteest, δ =. The pioneeing woks on appoximations to the uin pobability wee achieved by Camé and Lundbeg as ealy as the 193s unde the Camé-Lundbeg condition (Camé, 193). Howeve, thee ae many inteesting claim size distibutions that do not satisfy the Camé- Lundbeg condition. In paticula, Camé-Lundbeg appoximations ae not applicable fo few impotant medium and heavy tailed claim size distibution. The asymptotic behavios of the uin pobability in these cases ae totally diffeent fom those when the Camé- Lundbeg condition holds. Fo example, when the moment geneating function does not exist o a distibution is heavy tailed such as Paeto o lognomal distibutions, the Camé- Lundbeg condition is not valid. A eview of the asymptotic behavios of the uin pobability with medium and heavy tailed claim size distibutions ae available in Embechts et
2 2 Kume Pial Das al. (1997). This cuent study investigates a bound unde less stingent moment conditions and unde the assumption of non-zeo foce of inteest (δ ). The bound obtained in this study is bette fo the following easons. Fistly, this bound is applicable fo non-zeo foce of inteest. Most of the existing bounds have been developed fo zeo foce of inteest which is not common in pactice. Secondly, this bound can be used whee the classical Camé-Lundbeg is not applicable. And finally, this bound is tighte than that of Willmot (1994). 2 Definitions and Notation In isk theoy, the classical isk model is a compound Poisson isk model. Ruin theoy is consideed with the excess of the income (with espect to a potfolio of business) ove the outgo, o claims paid. This quantity, efeed to as the insue s suplus, may be expected to vay with time. Specifically uin is said to occu if the insue s suplus eaches a specified lowe bound. One measue of isk is the pobability of an event such as this, clealy eflecting the volatility inheent in the business. In addition, this seves as a useful tool in long ange planning fo the use of an insue s fund (Panje, 1992). Conside an insuance potfolio. Let U t denote the suplus of an insuance company at time t with u being the initial suplus. Suppose N t is the numbe of claims occuing in the potfolio in the time inteval (, t]. Assume that {N t, t } is a homogeneous Poisson pocess with intensity λ. Let T n denote the time of the nth claim with T =. Let Y i, i = 1, 2,... denote the claim sizes which ae identically independently distibuted positive andom vaiables with common distibution function F, whee F satisfies F () =. Also the claim amounts ae independent of the claim numbe pocess. The mean claim amount is denoted by µ = E[Y i ]. Let S t denote the accumulated amount of the claims occuing in the time inteval (, t], that is, S t = N t j=1 Y j. The aggegate pocess {S t : t },(with S t = if N t = ) is a compound Poisson pocess, specified by the paamete λ. The company eceives pemium income at ate c pe unit time. In addition to the pemium income, the company also eceives inteest on its eseves with the foce of inteest δ which is defined as δ = ln(1 + i) whee i is the effective ate of inteest. In theoy, the foce of inteest may vay instantaneously. Howeve, in pactice it is often a constant. Since the suplus at futue time is unknown, {U t } is a continuous-time stochastic pocess. Fom the peceding definitions, U t = ue δt + c e δ(t v) dv e δ(t v) ds v. (2.1) The concen hee is the event that uin occus, i.e., U t becomes negative fo some t. Thus,
3 Bounding The Ruin Pobability Unde Inteest Foce 3 the uin time is defined as { T = inf{t; U t < }, if U t < fo some t >, if U t fo all t >. Let ψ(u) be the uin pobability with initial suplus u when the foce of inteest is δ. This is the pobability that the suplus pocess of insue is below zeo at some time. Mathematically, ψ(u) = P (T < + ),that is, uin is said to occu at the fist instant the suplus becomes negative, if this eve happens. Since uin can occu only at the time of a claim, the uin pobability can also be defined as ( ψ(u) = P n=1 ) U Tn <. To avoid uin with cetainty o ψ(u) = 1, it is necessay to assume that the safety loading θ, defined by θ = c λµ λµ, is positive, o θ > which implies that c λµ >. The assumption c > λµ guaantees that fo any sample path, the suplus pocess difts ultimate to infinity, but the question is whethe it does this without eve cossing the hoizontal axis (time-axis). 3 Matingale Appoach The use of matingales in mathematical isk theoy is nealy as old as the matingale theoy itself, as can be ecognized by efeing to woks by de Finetti (1939) and by Täcklind (1942). The matingale technique has mostly been used to deive esults in connection with uin theoy, as shown in fundamental papes by Gebe (1973) and de Vylde (1977). Inteesting developments ae povided by Dalbean and Haezendock (1986). Moiconi (1986) shows how the sub-matingale decomposition theoem can be given an actuaial meaning. Moiconi examined the main consequences of the sub-matingale assumption in a discete time model and fomulated the uin theoy by classifying the gain pocesses accoding to the popeties of the set of the safety indexes of thei incements. Moiconi descibed a geneal isk model with an adjustable gain pocess and showed how it can be modified to obtain examples of embeddable sub matingales and P-sub matingales. Willmot and Yang (1996) have poved the classical Lundbeg s inequality by using a matingale technique. Søensen (1996) daws attention to a semi-matingale method which has been used to evaluate uin pobabilities. Howeve, Søensen used the inteest foce as zeo (δ = ) in that study. As fa as the pacticality concened it is ealistic to conside non zeo foce of inteest even though with non zeo inteest foce the mathematics become moe complicated. Vey ecently Das (29) has used a matingale and non-matingale appoach to find the uppe bound of the uin pobability. By definition, the pesent value of the suplus pocess {U t } is Z t = U t e δt. Thus,
4 4 Kume Pial Das fom Equation (2.1) Z t = u + c e δv dv e δv ds v. (3.1) Assume that c λµ > and define F as the σ-field geneated by the S pocess up to time. Fo t > [ ] [ ] E e δv ds v F = e δv ds v + E e δv ds v = e δv ds v + Now fom equation (3.1) and using the findings in equation (3.2), [ ] E[Z t F ] = u + c e δv dv E e δv ds v F = u + c = Z + (c λµ) e δv dv Z (since c > λµ). e δv dv e δv ds v λµ Thus, Z t is a sub matingale if c λµ >, which has been assumed. e δv λµ dv. (3.2) e δv dv + c e δv dv 3.1 Uppe bound using Dubins s Theoem Theoem 3.1. In the notation above the uin pobability ψ(u) satisfies ψ(u) c c + uδ povided the claim size andom vaiable Y has finite fist moment and c λµ >. Poof. Let [, u] be any non-empty eal inteval. The numbe of times the matingale {Z t : t < } move fom above u to below is known as the numbe of downcossings of the inteval [, u]. Let D u be the numbe of such downcossings. Dubins (1966) found a nice uppebound fo the numbe of upcossings of an inteval [, s] by a supe-matingale. By definition, Z t is a supe-matingale if and only if Z t is a sub-matingale, also an upcossing of [ u, ] fo Z t is a downcossing of [, u] fo Z t (it can be seen by eflecting the sample paths acoss the hoizontal axis). Thus, using a simila appoach to Dubins it s possible to have the following uppe bound fo the numbe of downcossings of an inteval [, u] by the sub-matingale Z t. Fo each k =, 1, 2,..., the numbe of downcossing, D u is a andom vaiable satisfying the following inequality P (D u > k) 1 (Z t u) + dp, (3.3) u D u=k
5 Bounding The Ruin Pobability Unde Inteest Foce 5 whee P is the pobability measue. In othe wods, the above integal is just an expectation. Moeove, (Z t u) + = sup(z t u, ). Summing ove k the inequality in (3.3) can be thought as a fom of Doob s esult which states that the expected value of D u is bounded fom above by 1 u (Zt u) +. The claims ente with a negative sign, so things only get lage without them. Thus fom (3.1), Z t u c e δv dv c δ (1 e δt ) as t. Fom the above inequality it can be claimed that (Z t u) is bounded above by c/δ. In othe wods, the pesent value (Z t u) is bounded by the pesent value of the pemium income to time infinity. Thus, the inequality (3.3) can be witten as P (D u > k) 1 u D u =k c δ dp c uδ P (Du = k). When the numbe of downcossings is zeo (k = ) the uin does not occu because the pocess does not move fom above u to below. Thus, P (D u = ) can be temed as the pobability of ultimate suvival 1 ψ(u). With this easoning the inequality (3.4) is equivalent to This completes the poof. ψ(u) c (1 ψ(u)) c uδ c + uδ. 4 Example: Paeto Claims In business and scietific models heavy-tailed o powe tailed distibutions ae often used to model catastophic o high consequence events and the Paeto is the simplest and most impotant heavy tailed distibution used (Ramsay, 29). The Paeto distibution is vey useful in modeling claim sizes in insuance, due in lage pat to its extemely thick tail. Fo example, in the line of fie insuance, the claim event is a fie in an insued stuctue that ceates a loss. Because fies cause heavy damage, adequate pobability should be assigned to the highe claim amounts. In actuaial liteatue, Bowes, Hickman, Gebe, Jones and Nesbit (1987) suggested Paeto distibution in such case. The Paeto distibution can be used to model any vaiable that has a minimum value and fo which the pobability density deceases geometically towads zeo. Conside the case when the claim sizes ae Paeto, i.e., αx α m f(x; x m, α) = x α+1, if x > x m >, α >, othewise.
6 6 Kume Pial Das It should be noted that in the actuaial liteatue, most of the studies employ not the classical Paeto distibution but the Paeto distibution of the above fom. Though mathematically simple, the Paeto distibution is vey difficult to wok and its Laplace tansfom is not easily manipulated. As a esult, eseaches have had to use appoximations to analyze stochstic models involving Paeto distibutions (Ramsay, 29). These appoximations include numeically inveting the Paeto s Laplace tansfom (e.g., Abate et al. 1999), discetization of the Paeto distibution( e.g., Dufesne and Gebe, 1989) o by a hype- Elang distibution (e.g., Wang et al. 25). Feldmann and Whitt (1998) have pointed out that thee ae many difficulties associated with using appoximations to detemine ight tail pobabilities due to Since the mean of Paeto does exist only fo α > 1, numeical illustations have been shown fo α = 2 and α = 1. In Figue 1 the bounds fo the uin pobabilities have been shown fo these two diffeent Paeto shape paametes. In both occassions, Paeto scale paamete has been consideed as x m = 2, foce of inteest has been chosen as δ =.5 and Poisson intensity has been chosen as λ = 1. With the choice of α = 2 the mean of Paeto claims is µ = αxm α 1 = 4. Since the assumption is c > λµ the pemium ate has to be lage than 4. Fo the illustation puposes c has been consideed as 5 fo this set of paametes. On the othe hand, fo α = 1 the pemium ate has been chosen as 2.5 since the mean claim size of a Paeto (1,2) distibution is appoximately Ruin Pobability Paeto(2,2) Paeto(1,2) Suplus (u) Figue 4.1: Ruin pobability fo Paeto claim sizes. Fom Figue 4.1 it can be obseved that the uin pobability with a highe aveage claim
7 Bounding The Ruin Pobability Unde Inteest Foce 7 size (Paeto (2,2)) is highe than that of the lowe aveage claim size (Paeto (1,2)). Even with the highe pemium ate Paeto (2,2) claim sizes have failed to poduce lowe isk than the Paeto (1,2) claim sizes. It is wothwhile to note that the Camé-Lundebeg bound does not exist fo the above mentioned two examples of Paeto distibution. 5 Compaing Two Uppe Bounds Willmot (1994) stated the following nice uppebound of uin pobability: ψ(u) E(Y 2 ) E(Y 2 ) + 2θE(Y )u, u. Hee u is the initial suplus, θ is the safety loading and claim amounts ae denoted by Y i with mean E(Y ) and vaiance E(Y 2 ) E(Y ) 2. The Willmot bound is applicable fo zeo inteest foce and the positive zeo inteest foce case seems to be quite a bit diffeent fom the zeo inteest foce case. Because when inteest foce is positive the pesent value of the pemium income out to infinity is finite (and unde some conditions, the pesent value of all futue claims is also finite). This seems to cause things to be quite a bit diffeent in the two cases. But since the Willmot bound is a tighte bound and an attempt has been made to compae the Willmot bound with the bound poposed in this study. Willmot s bound equie the existence of finite vaiance and that is why the above mentioned Paeto example discussed in Section 4 needs to be modified because vaiance is only defined fo α > 2 in Paeto distibution. Conside a Paeto (3,3) distibution with elative secuity loading θ =.5, Poisson intensity λ =.1 and foce of inteest δ =.5. Recall that Paeto (3,3) mean claim size is µ = 4.5. Fom the defintion of ealtive secuity loading it is possible to solve the peimum ate c as c = λµ(1 + θ). Fo the above paametes, the pemium ate has been calculated as c =.1 4.5(1 +.5) =.675. The following gaph (Figue 5.1) compaes the Willmot bound and the bound obtained in this study. It can be obseved fom Figue 5.1 that this poposed bound, at least in some cases, is shape than the Willmot bound. 6 Conclusion The esults obtained in this study ae impotant in the sense that these estimates of the uin pobability ae easy to calculate, shape than many existing bounds and can be used even when the moment geneating function does not exist. These less igoous moment conditions on the claim size andom vaiable ae impotant as fa as pactical use of these estimates is concened. Moeove, the foce of inteest in this study has been consideed as non-zeo while most of the othe studies conside a zeo foce of inteest.
8 8 Kume Pial Das R u i n P o b a b i l i t y Compaing Two Bounds Initial Suplus Poposed Bound Willmot Bound Figue 5.1: Compaing bounds of uin pobability. Acknowledgement The autho would like to thank the editos and the anonymous eviewes fo thei constuctive comments and suggestions to impove the quality of the aticle. The autho also thanks Pofesso Jey Veeh fo helpful convesations concening this poject. Refeences [1] J. Abate, G.L. Choudhuy and W. Whitt, (1999). An intoduction to numeical tansfom invesion and its application to pobability models, in: W. Gassman (ed.) Computational Pobability, pp Boston: Kluwe. [2] N. L. Bowes, J. C. Hickman, C. J. Nesbit, D. A. jones and H. U. Gebe, (1987). Actuaial Mathematics, Society of Actuay, [3] H. Camé, (193). On the Mathematical Theoy of Risk, Skandia Jubilee Volume, Stockhom. [4] K. P. Das, (29). Bounding the uin pobability: a matingale and a non-matingale appoach, Stochastic Analysis and Applications 27, [5] F. de Finetti, (1939). La teoia del ischio eil poblema della ovina dei giocatoi, Istituto Italiano degli Attuai 17, [6] F. Delbaen and J. Haezendonck, (1986). Matingales in Makov pocesses applied to isk theoy, Pesented at The 2nd NATO A.S.I on Insuance and Risk Theoy, 1985.
9 Bounding The Ruin Pobability Unde Inteest Foce 9 [7] F. de Vylde, (1977). Matingales and uin in a dynamic isk pocess, Scandinavian Actuaial Jounal 77, [8] L. Dubins, (1966). A note on upcossings of semi matingales, The Annals of Mathematical Statistics 37, 728. [9] F. Dufesne and H. U. Gebe, (1989). Thee methods to calculate the pobability of uin, ASTIN Bulletin 19, [1] P. Embechts, C. Klppelbeg and T. Mikosch, (1997). Modeling Extemal Events fo Insuance and Finance, Spinge, Belin. [11] A. Feldmann W. Whitt, (1998). Fitting mixtues of exponentials to long-tail distibutions to analyze netwok pefomance models, Pefomance Evaluation 3, [12] H. U. Gebe, (1973). Matingales in isk theoy, Mitteilun de Veeinigung schweceische Vesicheungsmathematike [13] H. U. Gebe, M. J. Goovaets and R. Kaas, (1987). On the pobability and seveity of uin, ASTIN Bulletin 17, [14] F. Moiconi, (1986). The Sub-matingale assumption in isk theoy, Insuance: Mathematics and Economics 5, [15] H. H. Panje and G. E. Willmot, (1992). Insuance Risk Models, Society of Actuaies, Schaumbug. [16] C. Ramsay, (29). The distibution of compound sums of Paeto losses, Scandinavian Actuaial Jounal 1, [17] M. Søensen, (1996). A semi-matingale appoach to some poblems in isk theoy, ASTIN Bulletin 26, [18] S. Täcklind, (1942). Su le isque de uine dans des jeux inéquitables, Skand. Aktuaietidskift 42, [19] J. Wang, H. Zhou, F. Xu and L. Li, (25). Hype-Elang based model fo netwok taffic appoximation, in: Y. Pan et al. (Eds), Paallel and distibuted pocessing and applications, pp , Spinge, Belin. [2] G. E. willmot, (1994). Refinements and distibutional genealizations of Lundbeg s inequality, Insuance: Mathematics and Economics 15, [21] G. E. Willmot and H. Yang, (1996). Matingales and uin pobability, Actuaial Reseach Cleaing House 1,
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